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Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics

  • Martín H. Escardó (a1)


We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the one-point compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot's and Ishihara's Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a well-foundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below εQ, and richer type systems allow us to get higher.



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[1] Barendregt, H. P., The lambda-calculus: its syntax and semantics, Studies in Logic and the Foundations of Mathematics, vol. 103, North-Holland, 1984.
[2] Beeson, M.J., Foundations of constructive mathematics, Springer, 1985.
[3] Bishop, E., Foundations of constructive analysis, McGraw-Hill Book Company, New York, 1967.
[4] Bove, A. and Dybjer, P., Dependent types at work, Language engineering and rigorous software development (Bove, A. et al., editors), Lecture Notes in Computer Science, vol. 5520, Springer, 2009, pp. 5799.
[5] Bridges, D. and Richman, F., Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.
[6] Bridges, D., Dalen, D. Van, and Ishihara, H., Ishihara's proof technique in constructive analysis, Indagationes Mathematicae, vol. 14 (2003), no. 2, pp. 163168.
[7] Bridges, D. and ViŢĂ, L., A general constructive proof technique, Electronic Notes in Theoretical Computer Science, vol. 120 (2005), pp. 3143.
[8] Coquand, T., Hancock, P., and Setzer, A., Ordinals in type theory, invited talk at Computer Science Logic, CSL 97,, 1997.
[9] Escardó, M. H., Infinite sets that admit fast exhaustive search, Logic in computer science, IEEE Computer Society, 2007, pp. 443452.
[10] Escardó, M. H., Exhaustible sets in higher-type computation, Logical Methods in Computer Science, vol. 4 (2008), no. 3, pp. 3:3, 37.
[11] Escardó, M. H., Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics, Martin-Löf formalization, in Agda notation, of part of the paper with the same title, University of Birmingham, UK,,2011.
[12] Escardó, M. H. and Oliva, P., Bar recursion and products of selection functions, available from the authors' web page, November, 2010.
[13] Escardó, M. H., Searchable sets, Dubuc-Penon compactness, omniscience principles, and the Drinker Paradox, Computability in Europe 2010 (Ferreira, F., Guerra, H., Mayordomo, E., and Rasga, J., editors), Centre for Applied Mathematics and Information Technology, Department of Mathematics, University of Azores, abstract and handout booklet, 2010, pp. 168177.
[14] Escardo, M. H., Selection functions, bar recursion and backward induction, Mathematical Structures in Computer Science, vol. 20 (2010).
[15] Grilliot, T. J., On effectively discontinuous type-2 objects, this Journal, vol. 36 (1971), pp. 245248.
[16] Hartley, J. P., Effective discontinuity and a characterisation of the superjump, this Journal, vol. 50 (1985), no. 2, pp. 349358.
[17] Ishihara, H., Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), no. 4, pp. 13491354.
[18] Escardo, M. H., Constructive reverse mathematics: compactness properties, From sets and types to topology and analysis (Crosilla, Laura and Schuster, Peter, editors), Oxford Logic Guides, vol. 48, Oxford University Press, Oxford, 2005, pp. 245267.
[19] Kreisel, G., Lacombe, D., and Shoenfield, J. R., Partial recursive functionals and effective operations, Constructivity in mathematics (Heyting, A., editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1959, pp. 290297.
[20] Palmgren, E., Constructivist and structuralist foundations: Bishop's and Lawvere's theories of, sets, Annals of Pure and Applied Logic, vol. 163 (2012), no. 10, pp. 13841399.
[21] Troelstra, A. S. and Dalen, D. Van, Constructivism in mathematics, Studies in Logic and the Foundations of Mathematics, vol. 121 and 123, North Holland, Amsterdam, 1988.
[22] Heijenoort, J. Van, From Frege to Godel: A source book in mathematical logic, 1879-1931, Harvard University Press, 1967.


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